Estimating Taylor SeriesCidTlli lfftif( )Consider a Taylor polynomial of a function f(x):)(,!))(()(nnafnaxafxTThen we find that the approximation stopping after n = k will be given by:(somewhat similar to that of the alternating series test)0n)!1())(()(1)1(kaxzfxRkkkWhere z is some (not usually known) number between a and x depending on howditifd Ihtth t R ( ) i thi dfmany derivatives were performed. In here, we note that Rk(x) is the remainder ofhow close f(x) is to the Taylor expansion stopping after k terms.Proving this theorem is a bit challenging, so we will not do so, but if we can place areasonable bound on z, we can show that the error is
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Estimating Taylor SeriesShdf( )T( )? Cidthf lliThSo when does f(x) = T(x)? Consider the following Theorem:Consider a Taylor Series of a function f(x):0)(,!))(()(nnnafnaxafxTThen T(x) = f(x) if and only if T(x) – f(x) = 0, if and only ifThen T(x)f(x) if and only if T(x)f(x)0, if and only if0)()(limxTxfkk0)(limxRkk0)!())((lim1)1(kaxzfkk1k